Eigenvalues and virtual levels of a family of $2\times 2$ operator matrices

In the present paper we consider a family of $2\times 2$ operator matrices ${\mathcal{A}}_{\mu }(k),$ $k\in {\mathbb{T}}^3:=(-\pi ,\pi {]}^3,$ $\mu >0,$ associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice ${\mathbb{Z}}^3,$ interacting via creation and annihilation operators. We prove that there is a value ${\mu }_0$ of the parameter $\mu$ such that only for $\mu ={\mu }_0$ the operator ${\mathcal A}\mu(\overline{0})$hasavirtuallevelatthepoint$z=0=\min\sigma{\rm ess}({\mathcal A}\mu(\overline{0}))$andtheoperator${\mathcal A}\mu(\overline{\pi})$hasavirtuallevelatthepoint$z=18=\max\sigma_{\rm ess}({\mathcal A}_\mu(\overline{\pi}))$,where$\overline{0}:=(0,0,0), \overline{\pi}:=(\pi,\pi,\pi) \in {\Bbb T}^3.$ The absence of the eigenvalues of ${\mathcal{A}}_{\mu }(k)$ for all values of $k$ under the assumption that $\mu ={\mu }_0$ is shown. The threshold energy expansions for the Fredholm determinant associated to ${\mathcal{A}}_{\mu }(k)$ are obtained.